Permanence and Global Attractivity of a Discrete Logistic Model with Impulses

Abstract

By piecewise Euler method, we construct a discrete logistic equation with impulses. The constructed model is more easily implemented at computer and is a better analogue of the continuous-time dynamic system. The dynamic behaviors of the constructed model are investigated. Sufficient conditions which guarantee the permanence and the global attractivity of positive solutions of the model are obtained. Numerical simulations show the feasibility of the main results.

1. Introduction

It is well known that impulsive differential equations have been considered by many authors (see, e.g., [1–7]). Such equations may exhibit several real world phenomena, such as rhythmical beating, merging of solutions, and noncontinuity of solutions. Since it is much richer than the corresponding theory of differential equations without impulsive effects, the theory of impulsive differential equations is emerging as an important area of investigation of model.

At all times, continuous-time dynamic systems play an important role in control theory, networks design, and so on. However, with the development of computer techniques, when implementing the continuous-time dynamic systems for computer simulation, for experimental or computational purposes, it is essential to formulate discrete-time dynamic systems which are an analogue of the continuous-time dynamic systems. These discrete-time systems, which are described by difference equations, inherit the similar dynamical characteristics. Because of that, more and more researchers pay their attention to the dynamical behaviors of difference equations (see [8–12]). However, few papers investigate the discrete model with impulses (see [13–17]). The main difficulty of constructing discrete model with impulses is how to describe impulsive moment. The main purpose of this paper is to construct the discrete model with impulses and investigate the permanence and global stability of the discrete model with impulses.

By piecewise Euler method, we construct the following discrete logistic model with impulses: 𝑥_{𝑚𝑘+𝑙+1}=𝑥_{𝑚𝑘+𝑙}𝑟𝑥exp1−^{𝑚𝑘+𝑙}𝐾,𝑥_𝑚𝑘+0=1+𝑏_𝑘𝑥_𝑚𝑘,𝑙=0,1,2,…,𝑚_𝑘+1−𝑚_𝑘−2,𝑚_𝑘+1−𝑚_𝑘−1,𝑘=0,1,2,…, (1) where the fixed moments of time 𝑚_𝑘 satisfy 𝑚₀=0, 𝑚_𝑘<𝑚_𝑘+1, and lim_𝑘→∞𝑚_𝑘=+∞. 𝑟 is the intrinsic rate and 𝐾 is the carrying capacity of the system; 𝑥₀=𝑥₀₊₀ is the initial value.

In this paper, we always assume that 0<𝐵₁<1+𝑏_𝑘<𝐵₂, 𝑘=1,2,3,….

We simply explain model (1).

It is obvious that 𝑥_{𝑚𝑘+𝑙}>0, 𝑘=0,1,2,…, 𝑙=0,1,2,…,(𝑚_𝑘+1−𝑚_𝑘−2),(𝑚_𝑘+1−𝑚_𝑘−1), for 𝑥₀>0.

When 𝑘=0, 𝑥_𝑙+1=𝑥_𝑙exp{𝑟(1−(𝑥_𝑙/𝐾))}, 𝑙=0,1,2,…,𝑚₁−1. We can obtain that 𝑥₁,𝑥₂,…,𝑥_𝑚1 and 𝑥_𝑚1+0=(1+𝑏₁)𝑥_𝑚1.

When 𝑘=1, 𝑥_𝑚1+𝑙+1=𝑥_𝑚1+𝑙exp{𝑟(1−(𝑥_𝑚1+𝑙/𝐾))}, 𝑙=0,1,2,…,𝑚₂−𝑚₁−1. We can obtain that 𝑥_𝑚1+1,𝑥_𝑚1+2,…,𝑥_𝑚2, and 𝑥_𝑚2+0=(1+𝑏₂)𝑥_𝑚2, and so on.

In some papers, authors use 𝑚⁺_𝑘 to denote impulsive moment (see [15]). It is obvious that describing the impulsive moment of model (1) is easily realized at computer. In addition, some authors use 𝑚_𝑘+1 to denote impulsive moment (see [13, 16]). Compared with it, model (1) is a better analogue of the continuous-time dynamic system.

The organization of this paper is as follows. In the next two sections, we give sufficient conditions on permanence and global stability of system (1), respectively. In Section 4, two examples are given. To conclude this paper, a discussion follows in Section 5.

2. Permanence

For convenience, let 𝑛=𝑚_𝑘+𝑙.

Definition 1. The species 𝑥_𝑛 of system (1) is said to be permanence if there exist positive constants 𝑀 and 𝑚 such that each positive solution 𝑥_𝑛 of the system satisfies 𝑚≤liminf_𝑛→∞𝑥_𝑛≤limsup_𝑛→∞𝑥_𝑛≤𝑀. (2) If the species of the system is permanence, then the system is called permanent.

Lemma 2. There exists a positive constant 𝑀, such that for every solution 𝑥_𝑛 of system (1), one has 𝑥_𝑛≤𝑀,𝑓𝑜𝑟𝑛>0. (3)

Proof. To prove (3), we have two cases.
Case I. For any 𝑥₀>0, we have 𝑥_𝑛+1=𝑥_𝑛𝑟𝑥exp1−^𝑛𝐾≤𝐾exp{𝑟−1}𝑟≜𝑀₁. (4) Here, we used max_𝑥≥0{={}𝑥exp𝑏−𝑎𝑥}exp𝑏−1𝑎. (5)
Case II. 𝑛=𝑚_𝑘+0, 𝑘=1,2,3,…. We have 𝑥_𝑚𝑘+0=1+𝑏_𝑘𝑥_𝑚𝑘=1+𝑏_𝑘𝑥_𝑚𝑘−1𝑟𝑥exp1−^𝑚𝑘−1𝐾≤𝐵₂𝐾exp{𝑟−1}𝑟≜𝑀₁. (6) Let 𝑀=max{𝑀₁,𝑀₁}. This completes the proof.

Lemma 3. Assume there exists a positive constant 𝐵₃, such that 𝐵₃<∏^∞_𝑘(1+𝑏_𝑘).Then there exist two positive constants 𝑚 and 𝑛₁, such that for every solution 𝑥_𝑛 of system (1), one has 𝑥_𝑛>𝑚 for 𝑛>𝑛₁.

Proof
Step I. Let 0<𝑚^′<𝐾. We will firstly prove that there exists 𝑛₁ such that 𝑥_𝑛1≥𝑚^′. Otherwise, for all 𝑛, 𝑥_𝑛<𝑚^′, 𝑥_𝑛+1=𝑥_𝑛𝑟𝑥exp1−^𝑛𝐾≥𝑥_𝑛𝑟𝑚exp1−^′𝐾. (7) Therefore, there exists a constant 𝑘^′, such that if 𝑘>𝑘^′, 𝑥_𝑚𝑘+1≥𝑥_{𝑚𝑘+1−1}𝑟𝑚exp1−^′𝐾≥𝑥_{𝑚𝑘+1−2}𝑚exp2𝑟1−^′𝐾⋮≥𝑥_𝑚𝑘+0exp𝑚_𝑘+1−𝑚_𝑘𝑟𝑚1−^′𝐾=1+𝑏_𝑘𝑥_𝑚𝑘exp𝑚_𝑘+1−𝑚_𝑘𝑟𝑚1−^′𝐾≥𝑥_𝑚𝑘′^𝑘_𝑖=𝑘′1+𝑏_𝑖exp𝑚_𝑘+1−𝑚_𝑘′𝑟𝑚1−^′𝐾⟶∞ (8) as 𝑘→∞, which contradicts the boundedness of 𝑥_𝑛. Hence, we conclude that there exists 𝑛₁ such that 𝑥_𝑛1≥𝑚^′.
Step II. If 𝑥_𝑛≥𝑚^′, for all 𝑛≥𝑛₁, then our aim is achieved. Otherwise, 𝑥_𝑛<𝑚^′ for some 𝑛>𝑛₁. Set 𝑛^∗=min_𝑛>𝑛1{𝑥_𝑛<𝑚^′}. We have 𝑥_𝑛≥𝑚^′ for 𝑛∈[𝑛₁,𝑛^∗).
Assume that 𝑛^∗∈[𝑚_𝑘,𝑚_𝑘+1). It is easy to see that 𝑥_𝑛∗<𝑚^′, 𝑥_𝑛∗−1≥𝑚^′.
There are two possible cases for 𝑛∈[𝑛^∗,𝑚_𝑘+1).
Case A. There exists a 𝑛^′′∈[𝑛^∗,𝑚_𝑘+1), such that 𝑥_𝑛′′≥𝑚^′. Let 𝑛=min_𝑛>𝑛∗{𝑥_𝑛≥𝑚^′}; then 𝑥_𝑛<𝑚^′ for 𝑛∈[𝑛^∗,𝑛) and 𝑥𝑛≥𝑚^′. Select an 𝑀 such that 𝑥_𝑛<𝑀,max1,𝐵₁𝑟𝑀exp1−𝐾<1. (9) If 𝑛^∗≠𝑚_𝑘+0, then 𝑥_𝑛∗=𝑥_𝑛∗−1𝑟𝑥exp1−^𝑛∗−1𝐾≥𝑚^′𝑟𝑀exp1−𝐾. (10) If 𝑛^∗=𝑚_𝑘+0, then 𝑚_𝑘≥𝑛₁.
Case 1 (𝑚_𝑘=𝑛₁). Consider 𝑥_𝑛∗=1+𝑏_𝑘𝑥_𝑚𝑘≥𝐵₁𝑚^′. (11)
Case 2 (𝑚_𝑘>𝑛₁). Consider 𝑥_𝑛∗=1+𝑏_𝑘𝑥_𝑚𝑘≥1+𝑏_𝑘𝑥_𝑚𝑘−1𝑟𝑥exp1−^𝑚𝑘−1𝐾≥1+𝑏_𝑘𝑚^′𝑟𝑀exp1−𝐾≥𝐵₁𝑚^′𝑟𝑀exp1−𝐾. (12) For 𝑛∈[𝑛^∗,𝑛−1), 𝑥_𝑛+1≥𝑥_𝑛𝑟𝑚exp1−^′𝐾⋮≥𝑥_𝑛∗exp𝑛+1−𝑛^∗𝑟𝑚1−^′𝐾. (13) If 𝑛^∗≠𝑚_𝑘+0, then 𝑥_𝑛+1≥𝑥_𝑛∗exp𝑛+1−𝑛^∗𝑟𝑚1−^′𝐾≥𝑚^′𝑟𝑀exp1−𝐾𝑟𝑚exp1−^′𝐾≥𝑚^′𝑟𝑀exp1−𝐾≜𝑚. (14) If 𝑛^∗=𝑚_𝑘+0, then 𝑥_𝑛+1≥𝑥_𝑛∗exp𝑛+1−𝑛^∗𝑟𝑚1−^′𝐾≥1+𝑏_𝑘𝑚^′𝑟𝑀exp1−𝐾×exp𝑛+1−𝑛^∗𝑟𝑚1−^′𝐾≥1+𝑏_𝑘𝑚^′𝑟𝑀exp1−𝐾𝑟𝑚exp1−^′𝐾≥𝐵₁𝑚^′𝑟𝑀exp1−𝐾≜𝑚. (15) The argument of Step II can be continued since 𝑥𝑛≥𝑚^′, and we have 𝑥_𝑛≥𝑚=min(𝑚,𝑚) for all 𝑛>𝑛₁.
Case B. 𝑥_𝑛<𝑚^′ for 𝑛∈[𝑛^∗,𝑚_𝑘+1). If 𝑥_𝑛≥𝑚^′ for all 𝑛≥𝑚_𝑘+1, then our aim is achieved. Otherwise, 𝑥_𝑛<𝑚^′ for some 𝑛>𝑚_𝑘+1. Set 𝑛=min_{𝑛>𝑚𝑘+1}{𝑥_𝑛<𝑚^′}.
For 𝑛>𝑛, the argument of step II can be continued since 𝑥_𝑛≥𝑚^′ for 𝑛∈[𝑚_𝑘+1,𝑛); hence 𝑥_𝑛≥𝑚=min(𝑚,𝑚), for all 𝑛>𝑚_𝑘+1. The proof is complete.

Combining Lemmas 2 and 3, we have proved the main result of this paper, which is stated next.

Theorem 4. If the condition of Lemma 3 holds, system (1) is permanent.

3. Global Attractivity of the Positive Solution

Next we discuss the global attractivity of the positive solution of system (1). In the following we say a positive solution of system (1) is globally asymptotically stable if it attracts all other positive solutions of the system.

Let 𝑥_𝑛=ln(𝑥_𝑛). Then system (1) becomes the following form: 𝑥_𝑛+1=𝑥_𝑛𝑒𝑥+𝑟1−^𝑛𝐾,𝑥_𝑚𝑘+0=ln1+𝑏_𝑘+𝑥_𝑚𝑘. (16)

Theorem 5. If 0<𝑀<2(𝐾/𝑟), then for any two positive solutions 𝑥_𝑛 and 𝑦_𝑛 of system (1), one has lim_𝑛→∞𝑥_𝑛−𝑦_𝑛=0. (17)

Proof. Let 𝑥_𝑛=ln(𝑥_𝑛) and let 𝑦_𝑛=ln(𝑦_𝑛). Then 𝑥_𝑛+1−𝑦_𝑛+1=𝑥_𝑛−𝑦_𝑛−𝑟𝐾𝑒𝑥_𝑛−𝑒𝑦_𝑛=𝑥_𝑛−𝑦_𝑛−𝑟𝐾𝑒^𝜉(𝑛)𝑥_𝑛−𝑦_𝑛=𝑟1−𝐾𝑒^𝜉(𝑛)𝑥_𝑛−𝑦_𝑛, (18) where 𝑒^𝜉(𝑛) is between 𝑥_𝑛 and 𝑦_𝑛 and 0<𝑚<𝑒^𝜉(𝑛)<𝑀 for sufficiently large 𝑛.
In view of 0<𝑀<2(𝐾/𝑟) such that 𝑟1−𝐾𝑒^𝜉(𝑛)<1, (19)
hence, lim_𝑛→∞𝑥_𝑛−𝑦_𝑛=0, (20) which implies that lim_𝑛→∞𝑥_𝑛−𝑦_𝑛=0. (21) This completes the proof.

4. Examples and Numerical Simulation

Example 1. Corresponding to the system (1), we assume that 𝑟=0.2,𝐾=100,𝑚₀𝑚=0,_𝑘=2^𝑘,𝑏_𝑘=sin𝑘,𝑘=1,2,…. (22) It is easy to see that the conditions in Theorem 4 are verified. Therefore, (1) is permanent (see Figure 1).

Figure 1 

Permanence of system (1) with initial condition 𝑥0=50.

Example 2. Corresponding to the system (1), we assume that 𝑟=0.2,𝐾=100,𝑚₀𝑚=0,_𝑘=2^𝑘,𝑏_𝑘=1+(−1)^𝑘1𝑘,𝑘=1,2,…. (23) It is easy to see that the conditions in Theorem 5 are verified. Therefore, (1) is global attractivity. Our numerical simulation supports our result (see Figure 2).

Figure 2 

Global attractivity of system (1) with initial conditions 𝑥0=50 and 𝑦0=60.

5. Conclusion

In this paper, by piecewise Euler method, we construct a discrete logistic equation with impulses. The model gives a new form of describing the impulsive moment, and the model is more easily implemented at computer and is a better analogue of the continuous-time dynamic system. The permanence and the global attractivity of positive solutions of the model are investigated. In our opinion, this discrete idea of the paper can apply to more complex model with impulses, such as model with impulses and delay and multigroup mode with impulses and delay. The other dynamic behaviors of discrete model can be researched, such as stability, periodic solution, and invariant sets.

Acknowledgment

This work is supported by the Key Projects in the National Science & Technology Pillar Program (no. 2011BAD37B01).